Firstly if you have any concerns about why you are being taught something or what your long term lesson plans are - ask your teacher?
Next there's a certain amount of your own research you can do yourself if you have no other teachers to fill the gap of what you are trying to accomplish.
Next set your own goals, and take them to your teacher and see if you can make a plan around accomplishing those goals, find out what your teacher can and can't offer you so you can work out what gaps you need to fill.
Piano lessons are not compulsory and neither are Piano teachers there's nothing wrong with having your own goals and using your teacher to get you there, rather than rely on your teacher to steer you and you question what you're doing (where you are now)
Lastly finger independence is a broad term and as a beginner i'm not sure it's what you're referring too. All hands can move independently, you typed this message on a keyboard right? one hand typed some letters, the other hand typed some more they're already independent of eachother it's just a new concept of playing the piano that makes you think you aren't.
Learning arpeggio chords is not wasted time. This is helpful to learning to play by ear. That is a useful skill but not the only one. I hope the teacher is teaching you the names of the chords you are learning, so you can relate them to charts in lead sheets and fake books.
....The task is difficult and boring the first year or two, but very rewarding in the third year and onwards.
(to be able to play pop/lounge music/video games music decently for myself).
you need to start out reading at least a little.He needs to address basic posture and fingering from the beginning. That might be more important than what you play.But you won't get it that cheap. 8 lessons a month for $25? most of us pay more than that for one lesson.
Can this teacher do what you want to be able to do? Do you enjoy his playing? It sounds like he has a good ear
is it because he plays by ear that you think he has no theory knowledge? Have you ever seen him read music?
He plays Maksim--are you Croatian? just curious--your English sounds very Eastern European --anyway...
who exactly has told you that you should be playing the piano not the keyboard
--who has put this idea of "proper lessons" in your head. I am guessing they are from the UK...or they speak English.. does this person who has told you all of this...do they play...or claim to play?
if this is your goal it sounds like you have the perfect teacher.
to play lounge music, video game music, pop standards... requires a solid ear far more than an ability to read large scores of piano music. If you want to play the Mario Brother's theme then you should not be studying with a professional classical teacher who is going to teach you Beethoven--you should be studying with someone who has a solid ear.
...except to say that this teacher you are with seems to be able to do--what you want to be able to do. Classical guys tend to struggle with jazz and improv in the same way the pop pianists tend to struggle with the classical pieces.
if you post a video of your playing and ask for comments on your technique you will get enough feedback to determine if you have any detrimental habits... then you can rest easy again.
He needs to address basic posture and fingering from the beginning. That might be more important than what you play.
You are wrong about not having access to other teachers. You made it to this forum so you have internet access. You can do Skype lessons with any country in the world.
$25 per month?? these must be group lessons, right?
I'm not a good at self teaching (for me: reading from internet counts as self teaching), for me I'm gonna need a routine face to face meeting with a teacher to progress. I guess I have nothing to worry about.